Be $V$ a real vector space of dimension $n+1$ defined over $\mathbb{R}^{n+1}$. Build up a differential structure on $\mathbb{P}(V)$, the projective space on $V$. Show that there is a diffeomorphism between $\mathbb{P}(V)$ and $G_{1}(V_{})$ (where $G_{1}(V_{})$ is the Grassmann manifold made up of vector subspaces of dimension $1$ ).
Definition 1 (Projective space $\mathbb{P}(V)$). Be $V$ a vector space over a field $\mathbb{K}$. The projective space $\mathbb{P}(V)$ of V is the set $\mathbb{P}(V)$ of the equivalence class of $V\setminus\{O\}$ with respect the the equivalence relation $\sim$ defined by $v\sim w$ iff $v=\lambda w$ for some $\lambda\in \mathbb{K}\setminus\{O\}$.
Definition 2 (Grassmann Manifold $G_{k}(V_{n})$) Be $V_{n}$ a vector space of dimension $n$ over a field $\mathbb{K}$, and be $k$, such that $1\leq k\leq n-1$. The Grassmann Manifold $G_{k}(V_{n})$ of the $k$planes of $V$ is the set $G_{k}(V_{n})$ of the vector subspaces of $V$ of dimension $k$.
The previous exercise is taken from the book: Marco Abate, Francesca Tovena, Geometria differenziale.
We have to find out a map $F: G_{1}(V)\rightarrow \mathbb{P}(V)$ such that $\phi\circ F\circ\chi^{-1}: \chi(W)\rightarrow \phi(U)$, is of $C^{\infty}$ where both $(\chi, W)$ and $(\phi, U)$ are charts for $G_{1}(V)$ and $\mathbb{P}(V)$ respectively.
Let be $V$ a vector space of dimension of $n+1$ over the field of real numbers. Once that a basis for $V$ has been fixed, it is possible to associate to each vector the coefficients with respect to the basis. Hence, for a generic vector $v\in V$, we have that its representation with respect to the basis is given by the following numbers $(x^{0},\dots,x^{n})$.
Considering $\mathbb{P}(V)$, we will indicate as $[x^{0}:\dots:x^{n}]$ the projection on $\mathbb{P}(V)$ of $(x^{0},\dots,x^{n})$.
The charts for $\mathbb{P}(V)$ are given by $$U_{j}=\{[x^{0}:\dots:x^{n}]\in\mathbb{P}(V)|x^{j}\neq0\}$$
with the transistion maps $\phi_{j}: U_{j}\rightarrow \mathbb{R}^{n}$ $$\phi_{j}([x^{0}:\dots:x^{n}]) = \Big(\frac{x^{0}}{x^{j}},\dots,\frac{x^{j-1}}{x^{j}},\frac{x^{j+1}}{x^{j}},\dots, \frac{x^{j+1}}{x^{j}}\Big)$$ and their inverse $$\phi^{-1}(y) = [y^{0}:\dots:y^{j-1}:1:y^{j+1}:y^{n}]$$.
The charts $(U_{0}, \phi_{0})$ and $(U_{1}, \phi_{1})$ are compatible: in fact $$\phi_{0}(U_{0}\cap U_{1})= \{y\in\mathbb{R}^{n}\ s.t. y^{1}\neq 0\} =\phi_{1}(U_{0}\cap U_{0})$$ and $$\phi_{0}\circ\phi^{-1}_{1}(y)=\big(\frac{1}{y^{1}},\frac{y^{2}}{y^1},\dots,\frac{y^{n}}{y^1}\big)$$. The same holds for the remaining charts. The topology of this manifold coincides with the quotient topology induced by $\mathbb{R}^{n+1}\setminus \{O\}$.
The charts of the Grassmann manifold $G_{k}(V)$ are defined as $$W_{Q}=\{P\in G_{k}(V)|P\cap Q = O\}$$ where $Q$ is the subspace of $V$ of dimension $n+1-k$ such that $V = Q\oplus P$. Let we fix an element $P_{0}$ of $W_{Q}$ and be $A\in Hom(P_{0},Q)$.
The charts $G_{k}(V)$ are $\psi_{Q}:A\rightarrow W_{Q}$ $$\psi_{Q}(A)= (\iota + A)(P_{0})$$
where $\iota:P_{0}\rightarrow V$ is the inclusion map. Since given $p\in W_{Q}$ every $p_{0}\in P_{0}$ can be decomposed in a unique way as $p_{0} = p +q$ with $p\in P$ and $q\in Q$. Hence $A$ can be defined in an explicit way as $A(p_{0})=-q$.
Be $\chi_{Q}=\psi^{-1}_{Q}$. Since fixing the basis for both $P_{0}$ and $Q_{0}$ is possible to identify $Hom(P_{0},Q)$ with $\mathbb{R}^{k\times (n+1-k)}$ the couples $(\chi_{Q},W_{Q})$ are $k\times(n+1-k)$ charts. It is possible to prove that they are compatible.
Regarding the topology, let $Y\subset \mathbb{R}^{k\times (n+1)}$ be the set of linearly independent n-tuples of elements of $\mathbb{R}^{k\times (n+1)}$. Then there is a surjection $p:Y\rightarrow G_{k}(V_{})$ sending an element of $Y$ to its span. The topology on $G_{k}(V_{})$ is the quotient topology for this map $p$ (considering $Y$ as a subspace of $\mathbb{R}^{k\times (n+1)}$)
Coming to the exercise, by setting $k=1$, we have that $G_{k}(V_{})$ is made up of $1-dimensional$ vector subspaces of $V$ (i.e. vectors of dimension $ n+1 $). Hence, we can define the map $F =(\pi\circ id_{\mathbb{R}^{n+1}})$ where:
$id_{\mathbb{R}^{n+1}}: V_{} \rightarrow \mathbb{R}^{n+1}$ is the canonical isomorphism between $V$, the vector spece defined over the field $\mathbb{R}^{n+1}$, and the field itself.
$\pi : V\setminus\{O\}\rightarrow \mathbb{P}(V)$ is the projection of $v\in V\setminus\{O\}$ onto $\mathbb{P}(V)$ i.e. $\pi : v\rightarrow [v]$
Since $F$ it is the composition of differential maps between manifolds (M.Abate, F.Tovena, Geometria differenziale pg 77), we have that $F$ it is a diffeomorphism.
Let me know what you think about that.
Thank you for your time.